A shadowable chain recurrent set with an attached hyperbolic singularity

Abstract

We prove that every factor map between topological flows preserves the standard shadowing property if it is injective except for a closed orbit that shrinks to a singularity. As an application, we construct a $C^{\infty }$-flow on a four-dimensional sphere whose nonwandering set contains an attached hyperbolic singularity yet possesses the standard shadowing property. This gives a counterexample to a conjecture given by Arbieto, López, Rego and Sánchez (Math. Annalen 390:417-437).

16 pages, 3 figures